Science:Math Exam Resources/Courses/MATH103/April 2013/Question 08 (c)
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Question 08 (c) 

Fundamental Theorem of Calculus Find 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

If you plug in the endpoint of the limit (that is, ) into the numerator (the integral) and denominator, what value do you get? What technique can you use to solve this? 
Hint 2 

Try L'Hopital's rule. 
Hint 3 

(Alternate Solution 2) Try to integrate the integral. 
Hint 4 

Use a substitution. 
Hint 5 

Use the substitution . 
Hint 6 

Then you can use L'Hopital's rule. 
Hint 7 

(Alternate Solution 3) Alternatively if you do not know L'Hopital's rule, you can use the Taylor series expansion for arctan. 
Hint 8 

(Alternate Solution 4) Try to figure out a Taylor series expansion for the integral and then use this to compute the limit. 
Hint 9 

Start with the expansion for and go from here. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Plugging in 0 into the integral yields
and plugging in 0 into also yields 0 hence we may apply L'Hopital's rule to see that
Notice that in the first equality, we also used the fundamental theorem of calculus. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let's try to evaluate the inside integral . Let so that , and . Plugging in yields . With the last equaltiy holding since plugging in 0 gives . Now our goal is to evaluate the limit . Plugging in zero into the numerator and denominator gives an indeterminant form 0/0 and so we may apply L'Hopital's rule to see that
completing the problem. 
Solution 3 

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Please rate my easiness! It's quick and helps everyone guide their studies. Proceed as in solution 2 to see that the integral evaluates to . Now we seek to evaluate
Taking the Taylor expansion of the function yields
As all terms bigger than when n equals 0 contain a positive power of x, taking the limit of the above as x tends to 0 leaves only the constant term which is

Solution 4 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. We start with the Taylor expansion for
Plugging in yields
Multiplying by gives
Now integrating from 0 to x gives
Dividing by gives
As all terms bigger than when n equals 0 contain a positive power of x, taking the limit of the above as x tends to 0 leaves only the constant term which is
which is the desired answer. 